THEORY AND APPLICATIONS OF FRACTAL TOPS MICHAEL BARNSLEY
Abstract. We consider an iterated function system (IFS) of one-to-one contractive maps on a compact metric space. We define the top of an IFS; define an associated symbolic dynamical system; present and explain a fast algorithm for computing the top; describe an example in one dimension with a rich history going back to work of A.Rényi [Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung.,8 (1957), pp. 477493]; and we show how tops may be used to help to model and render synthetic pictures in applications in computer graphics.
1. Introduction It is well-known that an iterated function system (IFS) of 1-1 contractive maps, mapping a compact metric space into itself, possesses a set attractor and various invariant measures. But it also possesses another type of invariant object which we call a top. One application of tops is to modelling and rendering new families of synthetic pictures in computer graphics. Another application is to information theory and data compression. Tops are mathematically fascinating because they have a rich symbolic dynamics structure, they support intricate Markov chains, and they provide examples of IFS with place-dependent probabilities in a regime where not much research has taken place. In these notes we (i) define the top of an IFS; (ii) define an associated symbolic dynamical system; (iii) present and explain a new fast algorithm for computing the top, based on the dynamical structure; (iv) describe an example in one dimension with a rich history; (v) explain and demonstrate that tops can be used to define and render beautiful pictures. This work was supported by the Australian Research Council. 2. The Top of an IFS Let an iterated function system (IFS) be denoted (2.1)
W := {X; w0 , ..., wN−1 }.
This consists of a finite of sequence of one-to-one contraction mappings (2.2)
wn : X → X, n = 0, 2, ..., N − 1
acting on the compact metric space (2.3)
(X, d)
with metric d so that for some (2.4)
0≤l 0 for the choice σ k = n, independent of all of the other choices. We also select X0 ∈ X and let Xn+1 = Wσn+1 (Xn ) for n = 0, 1, 2, ... .
Then, almost always, b lim {Xn : n = k, k + 1, ...} = A
k→∞
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3
where S denotes the closure of the set S. This algorithm provides in many cases a simple efficient fast method to compute approximations to the attractor of an IFS, for example when X = ¤, a compact subset of R2 . By keeping track of points which, for each approximate value of x ∈ X, have the greatest code space value, we can compute approximations to Gτ . We illustrate this approach in the following example which we continue in Section 6. Example 1. Consider the IFS (2.6)
{[0, 1] ⊂ R; w0 (x) = αx, w2 (x) = αx + (1 − α)}
We are interested in the case where 1 < α < 1, 2 which we refer to as "overlapping" because w0 ([0, 1]) ∩ w1 ([0, 1]) contains a nonempty open set. The two maps of the IFS are illustrated in Figure 1. In Figure 2
Figure 1. Graphs of the two transformations of the overlapping IFS in Example 1. See also Figure 2. **tifs1.gif b of the associated lifted IFS, and upon this attractor we have we show the attractor A indicated the top with some red squiggles. Figure 2 was computed using random iteration: we have represented points in code space by their binary expansions which are interpreted as points in [0, 1]. Since the invariant measure of both the IFS
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M ICHAEL BARNSLEY
Figure 2. The attractor of the IFS in Equation 2.7. This repreb of the lifted IFS corresponding to Equation sents the attractor A 2.6. The top of the IFS is indicated in red. The visible part of the "x-axis" represents the real interval [0, 1] and the visible part of the "y-axis" represents the code space Ω between the points 000000000.... and 111111111..... **graph2.gif and the lifted IFS contain no atoms, the information lost by this representation is irrelevant to pictures. Accordingly, the actual IFS used to compute Figure 2 is 1 1 1 (2.7) {[0, 1] × [0, 1] ⊂ R; W0 (x, y) = (αx, y), W2 (x, y) = (αx + (1 − α), y + )} 2 2 2 with α = 23 . 3. Application of Tops to Computer Graphics Here we introduce the application of tops to computer graphics. There is a great deal more to say about this, but to serve as motivation as well as to provide an excellent method for graphing fractal tops, we explain the basic idea here. A picture function is a mapping P : DP ⊂ R2 → C where C is a colour space, for example C =[0, 255]3 ⊂ R3 . The domain DP is typically a rectangular subset of R2 : we often take DP = ¤ := {(x, y) ∈ R2 : 0 ≤ x, y ≤ 1}. The domain of a picture function is an important part of its definition; for example a segment of a picture may be used to define a picture function. A picture in the usual sense may then be thought of as the graph of a picture function. But we will use the concepts of picture, picture function, and graph of a picture function
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interchangeably. We do not discuss here the important questions of how such functions arise in image science, for example, nor about the relationship between such abstract objects and real world pictures. Here we assume that given a picture function, we have some process by which we can render it to make pictures which may be printed, viewed on computer screens, etc. This is far from a simple matter in general. Let two IFS’s f := {¤; w W := {¤; w0 , ..., wN −1 } and W e0 , ..., w eN −1 }
and a picture function
e :¤→C P
e denote the attractor be given. Let A denote the attractor of the IFS W and let A f of the IFS W. Let τ :A→Ω denote the tops function for W. Let
e:Ω→A e⊂¤ φ
f Then we define a new picture denote the addressing function for the IFS W. function P :A → C by
e ◦ τ. e ◦φ P =P
f and the picture This is the unique picture function defined by the IFS’s W, W, e We say that it has been produced by tops + colour stealing. We think in P. e to "paint" code space; that is, this way: colours are "stolen" from the picture P e ◦φ e : Ω → C, which we then we make a code space picture, that is the function P use together with top of W to paint the attractor A. Notice the following points. (i) Picture functions have properties that are determined by their source; digital pictures of natural scenes such as clouds and sky, fields of flowers and grasses, seascapes, thick foliage, etc. all have their own distinctive palettes, relationships between colour and position, "continuity" and "discontinuity" properties, and so on. (ii) Addressing functions are continuous. (iii) Tops functions have their own special properties; for example they are continuous when the associated IFS is totally disconnected, and they contain the geometry of the underlying IFS attractor A plus much more, and so may have certain self-similarities and, assuming the IFS are built from low information content transformations such as similitudes, possess their own harmonies. Thus, the picture functions produced by tops plus colour stealing may define pictures which are interesting to look at, carrying a natural palette, possessing certain continuities and discontinuities, and also certain self-similarities. There is much more one could say here. e ◦φ e ◦ τ may be computed by random iteration, by coupling The stolen picture P f This is the method used until the lifted IFS associated with W to the IFS W. recently, and has been described in [3] and in [4]. Recently we have discovered a much faster algorithm for computing the entire stolen picture at a given resolution. It is based on a symbolic dynamical system associated with the top of W. We describe this new method in Section 5.
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In Figure 4 we illustrate the attractor, and invariant measure, and a picture defined by tops + colour stealing, all for the IFS of projective transformations an x + bn y + cn dn x + en y + fn (3.1) W = {¤; wn (x, y) = ( , ), n = 0, 1, 2, 3} gn x + hn y + jn gn x + hn y + jn where the coefficients are given by n an bn cn dn en fn gn hn jn 0 1.901 −0.072 0.186 0.015 1.69 0.028 0.563 −0.201 2.005 1 0.002 −0.044 0.075 0.003 −0.044 0.104 0.002 −0.088 0.154 2 0.965 −0.352 0.058 1.314 −0.065 −0.191 1.348 −0.307 0.075 3 −0.325 −0.0581 −0.029 −1.229 −0.001 0.199 −1.281 0.243 −0.058
The picture from which the colours were stolen is shown in Figure 3.
Figure 3. Colours were stolen from this picture to produce Figure 5 and the right-hand image in Figure 4. **starcolour2.bmp A close-up on the tops picture is illustrated in Figure 5.
Figure 4. From left to right, the attractor, an invariant measure, and a picture of the top made by colour stealing, for the IFS in Equation 3.1. Figure 5 shows a zoom on the picture of the top. **physics.bmp 4. The Tops Dynamical System We show how the IFS leads to a natural dynamical system T : Gτ → Gτ . The notation is the same as above. We also need the shift mapping defined by
S:Ω→Ω Sσ 1 σ2 σ 3 ... = σ 2 σ 3 ...
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Figure 5. Close-up on the fractal top in Figure 4. Details of the structure are revealed by colour-stealing. **physics6.gif for all σ = σ 1 σ 2 σ 3 .. ∈ Ω.
(x), Sσ) ∈ Gτ . Lemma 1. Let (x, σ) ∈ Gτ . Then (wσ−1 1 (x), Sσ) is uniquely defined and belongs to X × Ω because Proof. Notice that (wσ−1 1 (x), Sσ) ∈ / Gτ . Then there wn : X → X is one-to-one and S : Ω → Ω. Suppose (wσ−1 1 S −1 b b b it follows exists a point (wσ1 (x), ω) ∈ A where ω > Sσ. But since A = Wn (A) n
b But Wσ (wσ−1 (x), ω) = (x, σ 1 ω) and σ1 ω > σ, which is (x), ω) ∈ A. that Wσ1 (wσ−1 1 1 1 a contradiction. ¤
(y), Sω) = Lemma 2. Let (x, σ) ∈ Gτ . Then there is (y, ω) ∈ Gτ such that (wσ−1 1 (x, σ) ∈ Gτ .
Proof. Let y = (wN−1 (x), (N − 1)σ). Clearly (wσ−1 (y), S((N − 1)σ)) = (x, σ) and 1 b We just have to show (wN −1 (x), (N −1)σ) ∈ Gτ . Suppose (wN−1 (x), (N −1)σ) ∈ A. b with υ > (N − 1)σ. It follows that not. Then there exists a point (wN−1 (x), υ) ∈ A b it follows that υ 1 = (N − 1). It follows that Sυ > σ. Also since (wN −1 (x), υ) ∈ A −1 b and hence (x, Sυ) ∈ A. b It follows that (x, σ) ∈ / Gτ which WN −1 (wN −1 (x), υ) ∈ A is a contradiction. ¤ It follows that the mapping
T : Gτ → Gτ defined by T (x, σ) = (wσ−1 (x), Sσ) 1
is well-defined, and onto. It can be treated as a dynamical system which we refer to as {Gτ , T }. As such we may explore its invariant sets, invariant measures, other
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types of invariants such as entropies and information dimensions, and its ergodic properties, using "standard" terminology and machinery. We can we project {Gτ , T } onto the Ω-direction, as follows: let Ωγ := {σ ∈ Ω : (x, σ) ∈ Gτ for some x ∈ X} Then Ωγ is a shift invariant subspace of Ω, that is S : Ωγ → Ωγ with S(Ωγ ) = Ωγ , and we see that {Ωγ , S} is a symbolic dynamical system, see for example [9]. Indeed, {Ωγ , S} is the symbolic dynamical system corresponding to a partition of the domain of yet a third dynamical system {A, Te} corresponding to a mapping Te : A → A which is obtained by projecting {Gτ , T } onto A. This system is defined by −1 wN −1 (x) if x ∈ DN −1 := wN −1 (A), −1 (x) if x ∈ D w N −2 N −2 := wN−2 (A)\wN −1 (A) . . . (4.1) Te(x) = N−1 S w0−1 (x) if x ∈ D0 := w0 (A)\ wn (A) n=1
for all x ∈ A and we have
Te(A) = A.
We call {A, Te} the tops dynamical system (TDS) associated with the IFS. {Ωγ , S} is the symbolic dynamical system obtained by starting from the tops dynamical system {A, T } and partitioning A into the disjoint sets D0 , D1 , ..., DN −1 defined in Equation 4.1, where A=
N−1 [ n=0
Dn and Di ∩ Dj = ∅ for i 6= j.
An example of such a partition is illustrated in Figure 6.We refer to {Ωγ , S} as the symbolic tops dynamical system associated with the original dynamical system. A couple key phrases that relate to this type of situation are Markov Partitions and Specification Property. References include works by David Ruelle,Y. G. Sinai, Rufus Bowen, Caroline Series, Brian Marcus, and many others. Theorem 1. The tops dynamical system {A, Te} and the symbolic dynamical system {Ωγ , S} are conjugate. The identification between them is provided by the tops function τ : A → Ωγ . That is, Te(x) = φ ◦ S ◦ τ (x)
for all x ∈ A, and µ is an invariant measure for {A, Te} iff τ ◦ µ is an invariant measure for {Ωγ , S}.
Proof. Follows directly from everything we have said above.
¤
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Figure 6. Illustrates the domains D0 , D1 , D2 , D3 for the tops dynamical system associated with the IFS in Equation 3.1. This was the IFS used in Figures 4 and 5. Once this "picture" has been computed it is easy to compute the tops function. Just follow orbits of the tops dynamical system! **newleaftds.gif
5. Symbolic Dynamics Algorithm for Computing Tops Corollary 1. The value of τ (x) may be computed by following the orbit of x ∈ A as follows. Let x1 = x and xn+1 = Te(xn ) for n = 1, 2, ..., so that the orbit of x is {xn }∞ n=1 . Then τ (x) = σ where σ n ∈ {0, 1, ..., N − 1} is the unique index such that xn ∈ Dn for all n = 1, 2, ... Proof. This follows directly from Theorem 1. At the risk of being repetitive, notice c has totally disconnected attractor. Hence, the following points. (i) The IFS W b→A b which is equivalent to there is a well-defined shift dynamical system Tb : A b b S : Ω → Ω. A is the disjoint union of the sets Wn (A) and the code space address of b can be obtained by following its orbit under Tb and concatenating each point x ∈ A the indices of the successive regions Wn (A) in which it lands. The corollary follows because Te = Tb|Gτ . (ii) We may follow directly the orbit of x under Te and keep track of the indices as described. Use the strict contractivity of the maps to show that lim wσ1 ◦ wσ2 ◦ ...wσn (y) = x for all y ∈ A. The key observation is that the n→∞ orbit of a point on top stays on the top, and so the code given by its orbit is the value of the tops function applied to the point. ¤
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Corollary 1 provides us with a delightful algorithm for computing approximations to the top of an IFS in cases in which we are particularly interested, namely when the IFS acts in R2 as in Equation 3.1. Here we describe briefly one of many possible variants of the algorithm, with concentration on the key idea. (i) Fix a resolution L × M . Set up a rectangular array of little rectangular boxes, "pixels", corresponding to a rectangular region in R2 which contains the attractor of the IFS. Each box "contains" either a null value or a floating point Lm,n number xl,m ∈ R2 and a finite string of indexes ω l,m = σ1l,m σ2l,m ...σ l,m where Ll,m k denotes the length of the string and each σ l,m ∈ {0, 1, ..., N − 1}. The little boxes are initialized to null values. (ii) Run the standard random iteration algorithm applied to the IFS with appropriate choices of probabilities, for sufficiently many steps to ensure that it has "settled" on the attractor, then record in all of those boxes, which taken together correspond to a discretized version of the attractor, a representative floating point value of x ∈ A and the highest value, so far encountered, of the map index corresponding to that x-value. This requires that one runs the algorithm for sufficiently many steps that each box is visited many times and that, each time a map with a higher index value than the one recorded in a visited little box, the index value at the box is replaced by the higher index. [The result will be a discretized "picture" of the attractor, defined by the boxes with non null entries, partitioned into the domains D0 , D1 , .., DN −1 with a high resolution value of x ∈ A for each pixel. We will use these high resolution values to correct at each step the approximate orbits of Te : A → A which otherwise would rapidly loose precision and "leave" the attractor.] (iii) Choose a little rectangular box, indexed by say l1 , m1 , which does not have a null value. (If the value of τ (xl1 ,m1 ), namely the string ω l1 ,m1 , is already recorded in the little rectangular box to sufficient precision, that is Ll1 ,m1 = L, say go to another little rectangular box until one is found for which Ll1 ,m1 = 1.) Keep track of l1 , m1 , σ l1 ,m1 . Compute wσl1 ,m1 (xl1 ,m1 ) then discretize and identify the little box l2 , m2 to which it belongs. If Ll2 ,m2 = L then set ω l1 ,m1 = σ 1l1 ,m1 σ 1l2 ,m2 σ 2l2 ,m2 ...σ L−1 l2 ,m2 and go to (iv). If l2 , m2 = l1 , m1 set ω l1 ,m1 = σ 1l1 ,m1 σ 1l1 ,m1 σ 1l1 ,m1 ...σ 1l1 ,m1 and go to (iv). Otherwise, keep track of l1 , m1 , σ l1 ,m1 ; l2 , m2 , σ l2 ,m2 and repeat the iterative step now starting at l2 , m2 and computing wσl2 ,m2 (xl2 ,m2 ). Continue in this manner until either one lands in a box for which the string value is already of length L, in which case one back-tracks along the orbit one has been following, filling in all the string values up to length L, or until the sequence of visited boxes first includes the address of one box twice; i.e. a discretized periodic orbit is encountered. The strings of all of the points on the periodic orbit can now be filled-out to length L, and then the strings of all of the points leading to the periodic cycle can be deduced and entered into their boxes. (iv) Select systematically a new little rectangular box and repeat step (iii), and continue until all the strings ω l,m have length L. Our final approximation is τ (xl,m ) = ω l,m .
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This algorithm includes "pixel-chaining" as described in [8] and is very efficient because only one point lands in each pixel during stages (iii) and (iv). **Give an example here. 6. Analysis of Example 1 6.1. Invariant Measures and Random Iteration on Fractal Tops. We are interested in developing algorithms which are able to compute directly the graph Gτ of the tops function by some form of random iteration in which, at every step, the new points remain on Gτ . For while the just-described algorithm is delightful for computing approximations to the whole of Gτ it appears to be cumbersome and to have large memory requirements if very high resolution approximations to a part of Gτ are required, as when one "zooms in on" a fractal. But the standard chaos game algorithm has huge benefits in this regard and we would like to be able to repeat the process here. (The variant of the random iteration algorithm mentioned b and keeps track of "highest values" is earlier, where one works on the whole of A clearly inefficient in overlapping cases where the measure of the overlapping region is not negligible. Orbits of points may spend time wandering around deep within b rarely visiting the top!) In Section 7 use the insights gained in this section to A provide a random iteration algorithm which, for a class of cases, "stays on top". But this is not the only motivation for studying stochastic processes and invariant measures on tops. Such processes have very interesting connections to information theory and data compression. In the end one would like to come back, full-circle, to obtain insights into image compression by understanding these processes. This topic in turn relates to IFS’s with place-dependent probabilities and to studies concerning when such IFS’s possess unique invariant measures. Here we extend our discussion of Example 1 in some detail, to show the sort of thing we mean. We show that this example is related to a dynamical system studied by A. Renyi [10] connected to information theory. The IFS in this example also shows up in the context of Bernoulli convolutions, where the absolute continuity or otherwise of its invariant measures is discussed. We prove a theorem concerning this example which is, hopefully, new. Much of what we say may be generalized extensively. 6.2. The TIFS and Markov Process for Example 1. We continue Example 1. The tops IFS (TIFS) associated with the IFS in Equation 2.6 is the "IFS" made from the following two functions, one of which has as its domain a set that is not all of [0, 1]: (6.1)
1−α ); α w e1 (x) = αx + (1 − α) for all x ∈ [0, 1]. w e0 (x) = αx for all x ∈ [0,
These two functions are sketched in Figure 7. We are interested in invariant measures for the following type of Markov process. This relates to a "chaos game" with place-dependent probabilities. Define a Markov transition probability by (6.2)
Pe(x, B) = pe0 (x)χB (w e0 (x)) + pe1 (x)χB (w e1 (x)).
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Figure 7. The TIFS defined in Equation 6.1. This is the TIFS associated with the IFS illustrated in Figure 1. **tifs2.wmf Here (6.3) and (6.4)
pe0 (x) = pe1 (x) =
½
p0 0
f or f or
0 ≤ x < α1 − 1 1 α −1 0. Then µ e({a}) = pe0 (w e0−1 ({a}))e µ(w e0−1 ({a})) + pe1 (w e1−1 ({a}))e µ(w e1−1 ({a}))
from which it follows that either µ e({a}) = or or
p0 µ e(w e0−1 ({a})), −1 p1 µ e(w e1 ({a}))e µ(w e0−1 ({a})), −1 µ e(w e0 ({a})),
where the option is determined by the location of a. Equivalently 1 1 µ({a}) µ e({Te(a)}) = { , or , or1}e p0 p1
where the last option, µ e({Te(a)}) = µ e({a}), occurs iff Te(a) ≥ a. It follows that the mass of each succeeding point on the orbit of the point a under the TDS Te : [0, 1] → [0, 1] is greater than or equal to the mass of the point whose image under Te it is, with equality only when the succeeding point lies to the right of the point whose image it is. It follows that the orbit of a must be eventually periodic, for otherwise we have µ e([0, 1]) = ∞. If the orbit of a is eventually periodic, then without loss of generality we can take a to be a periodic point, and moreover can choose it to be the leftmost point on the orbit. Then if the orbit contains more than one point, the preimage of a on the orbit must lie to the right of a and consequently the measure of a must be strictly greater than the measure of a which is impossible. This leaves only the possibility that a = 1. ¤ 6.5. The TIFS and TDS for Example 1 in the trapping region. From this point forward in this section we concentrate on the behaviour of the TDS and the TIFS in the trapping region. We study invariant probability measures and ergodic properties for the TIFS and TDS restricted to the trapping region. In view of Theorem 2, we restrict attention to invariant probability measures which do not contain atoms. This will allow us to modify the "trapped" TDS/TIFS on any countable sets of points without altering potential invariant probability measures off a set of measure zero. Later, in Section 7, we will use the systems of functions and measures which we are now analyzing to construct new Markov processes on the top of the original IFS. We make the change of variable x0 = 1−α α x = g(x) to rescale the behaviour of the TDS restricted to Rtrapping to produce an equivalent dynamical system acting e e on [0, 1]: that is Te : [0, 1] → [0, 1] is defined by Te = g ◦ Te ◦ g −1 . The result is e e e 0 = [0, α), D e 1 = [α, 1] D
FRACTAL TOPS
and (6.6)
e Te(x) =
(
1 αx 1 αx
f or
− 1 f or
This is sketched in Figure 9. Notice that e (6.7) Te(x) = (βx)
15
ee x∈D 0 ee . x ∈ D1
where (y) denotes the fractional part of the real number y and 1 β= . α By using the symbol β we connect our problem to standard notation for a much studied dynamical system, Equation 6.7, see for example [10],[9]. The focus of all of the work, with which we are familiar, connected to the dynamical system in Equation 6.6, concerns invariant measures which are absolutely continuous with respect to Lebesgue measure and which maximize topological entropy. This is not our focus. We are interested in the existence and computation of invariant measures for the associated IFS with place-dependent, typically piecewise constant, probabilities.
Figure 9. The TDS in Figure 8 restricted to the trapping region and rescaled to [0, 1]. This trapped TDS is given by Equations 6.6. **tifs4.wmf
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Next we invert the trapped TDS to produce a corresponding restricted TIFS. This is defined with the aid of the two functions ee0 (x) = αx for all x ∈ [0, 1); (6.8) w ee1 (x) = αx + α for all x ∈ [0, 1 − α ]. w α These are sketched in Figure 10. The Markov process in the trapped region corre-
Figure 10. The trapped part of the rescaled TIFS, defined in Equation 6.8. This is the inverse of the dynamical system in Figure 9. **tifs5.wmf sponds to the transition probability e ee0 (x)) + e ee1 (x)). Pe(x, B) = e pe0 (x)χB (w pe1 (x)χB (w with the place-dependent probabilities ½ p0 f or e (6.9) pe0 (x) = 1 f or
and
(6.10)
e pe1 (x) =
½
p1 0
f or f or
0 ≤ x ≤ α1 − 1 1 α −1 0 and p0 + p1 = 1. Define a Markov transition probability by e ee0 (x)) + e ee1 (x)). pe1 (x)χB (w (6.11) Pe(x, B) = e pe0 (x)χB (w e pe1 (x) =
Then the Markov process possesses at least one and at most two linearly independent invariant probability measures. In particular, there exists at most two invariant probability measures which are also ergodic invariant measures for the tops dyname ical system Te : [0, 1] → [0, 1] defined by
1 e Te(x) = ( x) for all x ∈ [0, 1], α where (y) denotes the fractional part of the real number y. If α possesses certain arithmetic properties, namely that β = α1 is a "β-number", then the Markov process e possesses a unique invariant probability measure which is ergodic for Te. 6.7. First Part of the Proof : Corresponding Code Space Systems. We can convert the system in Theorem 3 to an equivalent symbolic dynamical system with the aid of the maps
w b0 (x, σ) = (αx, 0σ) for all (x, σ) ∈ [0, 1) × Ω 1−α w b1 (x, σ) = (αx + α, 1σ) for all (x, σ) ∈ [0, ]×Ω α We call the system described by Equation 6.12 "the lifted trapped TIFS"; it is illustrated in Figure 11. This system possesses a maximal invariant set Geτ ⊂ [0, 1) × Ω which obeys (6.12)
b0 (Geτ ) ∪ w b1 (Geτ ) Geτ = w
τ : [0, 1] → Ω. The projection of this system Geτ is the graph of a one-to-one function e onto the [0, 1]-direction gives us back the original system, the one in Theorem 3.
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Figure 11. The lifted trapped TIFS in Equation 6.12. The maximal invariant set, the attractor, is the graph of a function τ : [0, 1] → Ω which provides a symbolic dynamical system which e is equivalent to the original trapped TIFS. **tifs6.wmf Projection in the Ω-direction provides us with a symbolic dynamical system S : Ωγ → Ωγ where e τ ([0, 1]) = Ωγ , and S(Ωγ ) = Ωγ . {Ω, S} is the usual shift dynamical system on code space and the symbolic dynamical system {Ωγ , S} is obtained by restricting the domain of S to Ωγ . We could choose to write S : Ωγ → Ωγ as S|Ωγ : Ωγ → Ωγ and {Ωγ , S|Ωγ } = {Ωγ , S}, but do not. Note that we can compute τ (x) by following the orbit of x ∈ [0, 1] under the trapped TDS (Equation 6.6). e Thus we obtain the symbolic IFS with place-dependent probabilities {Ωγ ; s0 |Ωγ , s1 |Ωγ ; p0 (σ), p1 (σ)}
where (6.13)
s0 |Ωγ (σ) = 0σ for all σ ∈ Ωγ s1 |Ωγ (σ) = 1σ for all σ ∈ Ωγ ,
and the probabilities are given by pi (σ) = e pei (e τ −1 (σ)), that is ½ p0 for all σ ∈ Ωγ with 0 ≤ σ ≤ γ (6.14) p0 (σ) = 1 for all σ ∈ Ωγ with γ < σ ≤ 1
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and (6.15)
p1 (σ) =
½
p1 0
19
for all σ ∈ Ωγ with 0 ≤ σ ≤ γ for all σ ∈ Ωγ with γ < σ ≤ 1
where γ = e τ ( 1−α α ). This system is equivalent to the one in Theorem 3. It is the restriction to Ωγ of the IFS {Ω; s0 , s1 ; p0 (σ), p1 (σ)} where (6.16)
s0 (σ) = 0σ for all σ ∈ Ω s1 (σ) = 1σ for all σ ∈ Ω
and the probabilities pi : Ω → [0, 1] are defined in the same way as the pi : Ωγ → [0, 1] in Equations 6.14 and 6.15 with Ωγ replaced by Ω. It is helpful in some circumstances, such as when one needs to discuss the inverses of the maps in the IFS, to take the domain of s1 here to be Ω ∩ [0, γ] in place of Ω. Then the IFS is, strictly speaking, a local IFS, namely an IFS in which the domains of the functions need not be the whole space on which the IFS is defined. A similar remark applies to other IFS’s with place-dependent probabilities, when the probabilities are zero over parts of the underlying space. The system described by Equation 6.16 is an extension of the system described by Equation 6.13. Notice the following relationship between these two systems: Ωγ is the unique set attractor of the IFS {Ω; s0 , s1 |[0,γ] }; that is Ωγ is the unique nonempty compact subset of Ω such that Ωγ = s0 (Ω) ∪ s1 |[0,γ] (Ω) = s0 (Ω) ∪ s1 (Ω ∩ [0, γ]). By ignoring a countable set of points we can embed the code space in Ω in [0, 1] to make pictures of, and simplify the visualization of, the symbolic IFS’s in Equations 6.13 and 6.16. For example the symbolic system in Equation 6.13 may be represented by the following pretty IFS with place-dependent probabilities {[0, 1]; s0 : [0, 1] → [0, 1], s1 : [0, γ] → [0, 1]; p0 (x), p1 (x)} where (6.17)
s0 (x) = s1 (x) =
where p0 (x) = and p1 (x) =
½ ½
1 x for all x ∈ [0, 1] 2 1 1 x + for all x ∈ [0, 1] 2 2 p0 1
for all 0 ≤ x ≤ γ for all γ < x ≤ 1
p1 0
for all 0 ≤ x ≤ γ for all γ < x ≤ 1
Note that we use the symbols s0 (·) and s1 (·) to denote the maps and p0 (·) to denote the probabilities for embedded system. This embedded system is illustrated in Figure 12. Our Markov process, represented on Ωγ , involves applying the maps s0 |Ωγ , s1 |Ωγ with probabilities p0 , p1 respectively when σ ∈ Ωγ with σ ≤ γ, and applying the map s0 |Ωγ with probability one when σ ∈ Ωγ with σ > γ. This process equivalent e to the Markov process on [0, 1] corresponding to the transition probability Pe(x, B) in Equation 6.11 in Theorem 3.
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M ICHAEL BARNSLEY
Figure 12. The symbolic TIFS in Equation 6.17 which is (a) easyish to analyze and (b) holds the key to understanding all the earlier manifestations. The dotted line represents the portion of the graph of s1 in the region where it is never applied, i.e. p1 (x) = 0. **tifs7.wmf But we can also consider the corresponding process on Ω which involves applying the maps s0 , s1 with probabilities p0 , p1 respectively when σ ∈ Ω with σ ≤ γ, and applying the map s0 with probability one when σ ∈ Ω with σ > γ. This process extends, to all of Ω, the domain of the original symbolic process. We will find it very useful to consider this latter process. This is because, when we change α, the maps and the space upon which they act remain unaltered; only the set of values of σ ∈ Ω for which p1 (σ) = 0 changes. In the original system, in Theorem 3, the slopes of the maps, the location where a probability becomes zero, and the set of allowed codes, all change with α. 6.8. Transfer operator and Probability operator for the system in Theorem 3. We will find that the structure of the system in Theorem 3 depends fundamentally on whether or not γ, the address of the point 1−α α , which may be computed by following the orbit of 1−α under the trapped tops dynamical system α ee T : [0, 1] → [0, 1], terminates in an endless string of zeros or not. That is, on whether or not r 1−α = =β−1 2k−1 α
FRACTAL TOPS
21
for some pair of positive integers k and r. We work with the closure of the symbolic system. We are interested in those invariant measures which correspond to orbits of the following type of random iteration. When the current point lies in the region 0000... ≤ σ ≤ γ there is a nonzero probability that s0 may be applied to the current point and there is a nonzero probability that s1 may be applied to the current point. When the current point lies in the region γ < σ < 1 the map s0 is applied with probability one. Theorem 2 implies that this Markov process possesses no invariant measures which include point masses. Notice that if γ terminates in 0 or 1 then S : Ωγ → Ωγ is open, that is it maps open sets to open sets. This implies that the inverse branches of S are continuous in the product topology. Let X ∈ {[0, 1], Ω, Ωγ }. We need the transfer operator U : L0 → L0 , where 0 L = L0 (X) is the space of bounded Borel measurable functions on X, defined by (U f )(·) :=
1 X i=0
pi (·)f (ti (·)) for all f ∈ L0 .
eei , or si , or si |Ω , according as X =[0, 1], or Ω, or Ωγ , respectively. We where ti = w γ also need the adjoint operator U ∗ : P → P, where P = P(X), the space of all Borel probability measures on X. U ∗ is defined by Z Z ∗ ∗ (6.18) (U υ)(f ) = f d(U υ) := U f dυ for all f ∈ L0 and υ ∈ P. X
X
∗
We refer to U as the probability operator associated with the corresponding IFS. We interpret these operators U and U ∗ as acting with underlying space Ω, Ωγ or [0, 1] according to context. 6.9. Example : The case γ = 110. At this point, to illustrate some key ideas, we look at a simple example of the case where γ terminates in 0, namely γ = 110. We modify the Markov process using the symbolic IFS by replacing ≤ γ by < γ, that is: when σ
< 110 apply maps s0 , s1 with probabilities p0 , p1 respectively otherwise apply map s0
Look at what happens on the four cylinder sets C00 , C01 , C10 , C11 . What does the operator U ∗ do to a measure ν ∈ P(Ω)? Let us write ν = ν 00 + ν 01 + ν 10 + ν 11
where ν ij is zero off the cylinder Cij , to denote the four obvious components of ν. Then the new measure e ν = U ∗ ν on C00 is given by ν 00 (B) = p0 ν 00 (SB) + p0 ν 01 (SB), e
for all Borel subsets B ∈ B(Ω). It consists of two components, one supported on the cylinder C000 ⊂ C00 and one supported on the cylinder C001 ⊂ C00 . We also find, using similar notation, that νe01 (B) = p0 ν 10 (SB) + ν 11 (SB),
ν 10 (B) = p1 ν 00 (SB) + p1 ν 01 (SB), e ν 11 (B) = p1 ν 10 (SB). e
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M ICHAEL BARNSLEY
See Figure 13.
Figure 13. Illustrates the redistribution of mass between cylinder sets by the probability operator in Example 6.9. **bins1.gif In particular, if the measure ν is invariant, then the masses of each of the cylinders must be preserved, which implies ν(C00 ) ν(C00 ) p0 p0 0 0 0 0 p0 1 ν(C01 ) ν(C01 ) (6.19) p1 p1 0 0 ν(C10 ) = ν(C10 ) . 0 0 p1 0 ν(C11 ) ν(C11 )
The matrix here is the transpose of an irreducible stochastic matrix and by wellknown theorems [**] Equation 6.19 possesses a unique solution, an eigenvector with strictly positive entries, with eigenvalue one, with total mass one. In fact, in this case, we readily find: (6.20)
ν(C00 ) =
p0 p1 p1 p2 , ν(C01 ) = , ν(C10 ) = , ν(C11 ) = 1 c c c c
where c = 1 + p1 + p21 . Now let e P(Ω)= {ν ∈ P(Ω) : ν obeys Equation 6.20. e Then ( P(Ω), dMK ) is a complete metric space, where dMK denotes the MongeKantorovitch metric. e e By what we have just proved, U ∗ : P(Ω)→ P(Ω). Furthermore, we claim that dM −K (U ∗ ν, U ∗ µ) ≤ ldM−K (ν, µ).
FRACTAL TOPS
23
To see that this is true, and much more as well, we set up some notation and a framework. 6.10. A general framework. Let {Ki : i = 0, 1, 2, ..., N − 1} be a finite sequence of disjoint compact metric spaces. The metric on each is denoted by d = di . Let K =K0 ∪ K1 ∪ ... ∪ KN −1 . Define a metric d∪ on X by ½ d(x, y) if x, y ∈ Ki for some i ∈ 0, 1, ..., N − 1, d∪ (x, y) = D if x ∈ Ki and y ∈ Kj for some i 6= j, with i, j ∈ 0, 1, ..., N − 1, where D > 0 is a constant. Then (K, d∪ ) is a complete metric space. Let wij : Ki → Kj be a one-to-one Lipshitz function with Lipshitz constant lij , that is d(wij (x), wij (y)) ≤ lij d(x, y) for all x, y ∈ Ki for all i, j = 0, 1, ..., N − 1. Use these functions to define new functions Wi : K → K according to Wi (x) = wij (x) when x ∈ Ki , i = 0, 1, ..., N − 1. Notice that Wi is continuous because ½ ¾ d∪ (Wi (x), Wi (y)) ≤ max lij d∪ (x, y) whenever d∪ (x, y) < D. j
Let (pij ) denote a stochastic matrix of transition probabilities. That is, we assume that pij ≥ 0, where pij is thought of as "the probability of transition from space Ki to space Kj ", and we assume N−1 X
pij = 1.
j=0
Let m0 , m1 , ..., mN −1 denote an eigenvector such that mn > 0, m0 + m1 + ... + mN −1 = 1, and N −1 X i=0
mi pij = mj for j = 0, 1, ..., N − 1,
corresponding to eigenvalue one. At least one such eigenvector exists. This eigenvector represents a stationary probability distribution for a Markov process which is represented by the stochastic matrix. Let e P(K) = {ν ∈ P(K) : ν i (Ki ) = mi }.
Now notice that, for any set of numbers m0 , m1 , ..., mN−1 with mn > 0 for all n, e (P(K), dMK ) is a complete metric space, where ν i ∈ M(Ki ) denotes the restriction
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M ICHAEL BARNSLEY
e e of ν ∈ P(K) to Ki . The Monge-Kantorovitch metric on P(K) and P(K) is defined by Z dMK (ν, µ) = max f (x)(dν(x) − dµ(x)) f ∈Lip1 (K)
=
max
f ∈Lip1 (K)
K
N −1 X i=0
Z { f |Ki (xi )(dν i (xi ) − dµi (xi )} Ki
where Lip1 (K) ={f : K → R : |f (x) − f (y)| ≤ d∪ (x, y) for all x, y ∈ K}. Let place-dependent, piecewise constant probabilities be defined on K by pj (x) = pij when x ∈ Ki so that N −1 X j=0
pj (x) = 1 for each x ∈ K.
We consider the Markov process on K with transition probability P (x, B) =
N −1 X
pj (x)χB (Wj (x)).
j=0
e e →P(K) where The corresponding probability operator is U ∗ : P(K) Z −1 N−1 X NX −1 P (x, B)dU ∗ ν(x) = pij ν(wij (B)) for all B ∈ B(K). (U ∗ ν)(B) = K
i=0 j=0
Notice that the range of U ∗ is indeed contained in eP(K) because −1 N−1 X NX
∗
(U ν)(Kk ) =
−1 pij ν(wij (Kk ))
i=0 j=0
N−1 X
=
=
N−1 X
−1 pik ν(wik (Kk ))
i=0
N −1 X
pik ν(Ki ) =
i=0
pik mi = mk .
i=0
Let f : K → R be Borel measurable and let us denote the restriction of f to Ki by fi . Then the transfer operator is given by (U f )(x) =
N−1 X
pj (x)f (Wj (x));
j=0
the restriction of U f to Ki is
(U f )|Ki (x) =
N −1 X
pij f |Kj (wij (x));
Z
f (x)d(U ∗ v)(x).
j=0
and we have
Z
K
U f (x)dv(x) =
X
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25
Theorem 4. Let D ≥ max max di (x, y) i
x,y∈Ki
and Li :=
N −1 X j=0
pij lij > 0 for i = 0, 1, ..., N − 1.
When L := max i
N −1 X
pij lij < 1
j=0
e e →P(K) is a contraction mapping with contractivity factor the operator U ∗ : P(K) L and possesses a unique fixed point. That is, the Markov process has exactly one invariant probability measure µ such that µ(Ki ) = mi for i = 0, 1, ..., N − 1. Proof. (6.21)
dMK (U ∗ υ, U ∗ µ) Z = max U f (x)(dν(x) − dµ(x)) f ∈Lip1 (K)
=
=
=
max
f ∈Lip1 (K)
max
f ∈Lip1 (K)
max
f ∈Lip1 (K)
where
has the property that
K
N −1 Z X
i=0 K i
(U f )|Ki (xi )(dν i (xi ) − dµi (xi ))
N −1 Z N −1 X X i=0 K j=0 i
N −1 X i=0
Li
Z
Ki
pij f |Kj (wij (xi ))(dν i (xi ) − dµi (xi ))
fe|Ki (xi )(dν i (xi ) − dµi (xi ))
N −1 1 X e pij f |Kj (wij (xi )); f |Ki (xi ) = Li j=0
|fe|Ki (xi ) − fe|Ki (yi )| ≤ d(xi , yi ) for all xi , yi ∈ Ki .
Now notice that we can alter the value of fe|Ki by a constant without altering e the value of the last expression in Equation 6.21. So we define fe : K → R by maxx∈Ki fe(x) − minx∈Ki fe(x) e fe(x) = fe(x) − when x ∈ Ki , for i = 0, 1, ..., N − 1, 2
and we choose D ≥ maxi maxx,y∈Ki di (x, y). Then the distance between distinct Ki ’s is greater than the variation of fe and it follows that e e |fe(x) − fe(y)| ≤ d(x, y) for all x, y ∈ K.
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M ICHAEL BARNSLEY
Hence, ∗
∗
dMK (U υ, U µ) = L = L ≤ L
max
N −1 Z X
fe∈Lip1 (K) i=0 Ki
max
fe∈Lip1 (K)
max
e fe∈Lip1 (K)
Z
ZK K
e fe|Ki (xi )(dν i (xi ) − dµi (xi ))
e fe(x)(dν(x) − dµ(x))d(ν, µ)
e fe(x)(dν(x) − dµ(x))d(ν, µ) = LdMK (ν, µ).
¤
6.11. Completion of the proof of Theorem 3 in the case γ = 110. We simply apply Theorem 4 to the Example in Section 6.9. In Example 6.9 we have N = 4, the transition matrix pij is the transpose of the one in Equation 6.19, namely p0 0 p1 0 p0 0 p1 0 0 p0 0 p1 , 0 1 0 0
and possesses a unique stationary probability distribution, and L = 0.5. It follows from Theorem 4 that the corresponding symbolic IFS with place-dependent probabilities possesses a unique invariant measure. This measure is non-atomic. It can be mapped to an invariant measure for the corresponding trapped TIFS.
6.12. Remark. In Theorem 4, in place of the space K, we could have used the space e K=K 0 × K1 × ... × KN−1 . Then we would have defined ν ∈ P(m0 ,m1 ,...,mN−1 ) := Pm0 (K0 ) × Pm1 (K1 ) × ... × PmK−1 (KN−1 ) where Pmi (Ki ) denotes the set of positive finite Borel measures on Ki such that υ i ∈ Ki implies υ(Ki ) = mi . In this case we would write ν = (v0 , v1 , ..., vN−1 ). We would then have defined a metric on P(m0 ,m1 ,...,mN −1 ) according to: Z N−1 X dM K2 (ν, µ) = max { fi (xi )(dν i (xi ) − dµi (xi )} i=0
fi ∈Lip1 (Ki )
Ki
This would have provided us with the complete metric space (P(m0 ,m1 ,...,mN−1 ) , dMK2 ) and we could have developed our results in this space. Such an approach would have avoided reference to d∪ and has certain advantages and certain disadvantages.(**) The conclusions regarding contractivity of the probability operator using this alternative framework would have been essentially the same as we obtained in Theorem 4. 6.13. Completion of the proof of Theorem 3 in the cases γ = x....xx10 and γ = x....xx01.. We claim that all cases where γ = γ 1 ....γ n−1 10 work somewhat similarly to the case γ = 110. The main points to establish are: (i) There exists a finite "core" collection of cylinders {Ciγ ⊂ Ω : i = 1, 2, ..., m(γ)} such that for each i ∈ {1, 2, ..., m(γ)} there are j, k ∈ {1, 2, ..., m(γ)} such that s00 (Ciγ ) = s0 (Ciγ ) ⊂ Cjγ and s01 (Ciγ ) := s1 |[0,γ] (Ciγ ) = (s1 ([0, γ] ∩ Ciγ ) ⊂ Ckγ .
FRACTAL TOPS
27
where s00 denotes the mapping s0 : Ω → Ω and s01 denotes the mapping s1 |[0,γ] , defined to act on subsets of Ω according to s01 (B) = (s1 ([0, γ] ∩ B) for all B ⊂ Ω. (ii) Let σ 1 σ2 ...σ n ∈ {0, 1}n . Then s0σ1 ◦ s0σ2 ◦ ... ◦ s0σn (Ω) =
½
either ∅ or Ciγ
for some i ∈ {1, 2, ..., m(γ)}. (iii) The Markov process restricted to the "core" cylinders, is recurrent, or regionally transitive, that is, that one can go from cylinder to cylinder in the core with finite probability in finitely many steps. Some detail is needed to show these points. This involves direct analysis of the transition matrix n−1 (pi,j )2i,j=0 where pi,j = probability of transition from Cσ1 σ2 ...σn to Cσe 1 σe 2 ...eσn
where i = σ 1 + 2 · σ 2 + ...2n−1 σ n and j = σ e1 + 2 · σ f2 + ...2n−1 σ en . In this regard we note the following: pl,[l/2] = p0 , and pl,2n−1 +[l/2] = p1 for all l = 0, 1..., lγ ,
pl,[l/2] = 1 and pl,2n−1 +[l/2] = 0 for all l = lγ + 1, ..., 2n − 1, where lγ := γ 1 + 2 · γ 2 ....2n−2 · γ n−1 + 2n−1 , and that pi,j = 0 for all values of i, j other than those for which pi,j has already been defined. These values enable us to explicitly demonstrate our assertions regarding eventual regional transitivity and uniqueness of the stationary state of the stochastic matrix. For example, when γ = 1010 we find p0 0 0 0 p1 0 0 0 p0 0 0 0 p1 0 0 0 0 p0 0 0 0 p1 0 0 0 p0 0 0 0 p1 0 0 . (pi,j ) = 0 0 p0 0 0 0 p1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 m(γ)
It follows from (i), (ii) and (iii) that: (iv) If υ ∈ P(Ω) then (U ∗ )n υ ∈ P(∪i Ciγ ). m(γ) γ γ ∗ Ci ) and υ(Ci ) is uniquely determined. Indeed, (v) If U υ = υ then υ ∈ P(∪i υ(Ciγ ) = υ i for i = 1, 2, ..., m(γ) where υ i is the stationary probability distribution e m(γ) C γ ) denote the space of for the corresponding stochastic matrix. Now let P(∪ i i m(γ) γ Ci such that µ(Ciγ ) = υ i . Then as in Theorem 4 probability measures µ on ∪i e m(γ) C γ ) → P(∪ e m(γ) C γ ) is a strict contraction, which provides the rest of U ∗ : P(∪ i i i i the proof of Theorem 3 in the case γ = x....xx10. The case γ = xxx01 is just like the case γ = xxx10 after some possible modification on a set of measure zero.
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M ICHAEL BARNSLEY
6.14. Another approach to the proof of Theorem 3 in the case γ = x....xx10 using the formulation and a theorem of Parry. Another approach makes connection to the work of W. Parry [9]: in this approach we quote Parry’s work to show that the underlying structure is that of an intrinsic Markov chain. However our framework, above, is broader since we work on all of Ω, which has a further implication: when we consider associated ergodic theorems, which we do not do in these notes at this time, in the "recurrent" case, we obtain a stronger result than I. Werner, [11], regarding convergence of orbits produced by random iteration in the case of certain graph-directed IFS. This also has implications which are entirely new, so far as we know, for the "not open" case, which corresponds for example to the case where γ is irrational, which we discuss in later sections. **Recurrence of the symbolic system {Ωγ , S|Ωγ } follows from the fact that the e corresponding restricted tops dynamical system Te : [0, 1] → [0, 1] is strictly expande ing and, if O ⊂ [0, 1] is nonempty and open, then Te(O) = [0, 1] for sufficiently large integer n. **The symbolic system {Ωγ , S|Ωγ } here is exactly the one considered by W.Parry [9]. By combining Parry’s set-up with piecewise-constant probabilities we obtain a framework to which Theorem 4 applies, with L = 0.5. *Ωγ has the compact metric totally disconnected product topology which arises from the product topology on Ω. * Cylinders in Ωγ are cylinders in Ω intersected with Ωγ . Cylinders are both open and closed in the relative topology. *{Ωγ , S} is an intrinsic Markov chain of order r iff Ωγ ∩ Cσ0 σ1 ...σn 6= ∅ with n ≥ r and Ωγ ∩ Cσn−r+1 ...σn σn+1 6= ∅ imply Ωγ ∩ Cσ0 σ1 ...σn+1 6= ∅. * Regional transitivity is defined as follows: (i) given non-empty cylinders C, D ◦n of Ωγ there exists n such that S ◦n C ∩ D 6= ∅ and (ii) ∪∞ n=0 S C = Ωγ . These two conditions are equivalent when S : Ωγ → Ωγ is open. We have extracted the following theorem from the first few pages of [9]. The second sentence in our statement is contained in Parry’s proof of the first sentence. Theorem 5. [9] The symbolic dynamical system {Ωγ , S} is an intrinsic Markov chain iff S : Ωγ → Ωγ is open. In this case Ωγ can be partitioned into a set of non-empty cylinders Ωγ ∩ Cσ1 σ2 ...σn , each of the same length n, such that S(Ωγ ∩ Cσ1 σ2 ...σn ) = Ωγ ∩ Cσ2 ...σn . 6.15. Continuation of the proof of Theorem 3 in the case where the expansion of γ does not terminate in 0 or 1. Let γ = γ 1 γ 2 ... not terminate in 0 or 1. Let µ(n) ∈ P(Ω) denote the unique invariant measure corresponding to γ (n) = γ 1 ...γ n−1 10. Notice that γ = lim γ (n) . Let µ b denote a limit point of n→∞
{µ(n) : n = 1, 2, ...}. Then we prove that µ b is an invariant measure for the process corresponding to that value of γ. Let µ b = lim µ(nm ) where where , and {µ(nm ) }∞ m=1 is an infinite subsequence nm →∞
∗ of {µ(n) }∞ b=µ b. **We seem n=1 . Then we begin by showing that showing that Uγ µ (n) to be using two notations: µ = µγ (n) . (n)
We need some more notation. For each positive integer n, C γ denotes the unique 2
cylinder of length n which contains the infinite string
γ 2.
(n)
C γ + 1 denotes the unique 2
2
FRACTAL TOPS
29
cylinder of length n which contains the infinite string γ2 + 12 . Notice that neither (n) of these strings terminates in 0 or 1. Let Cm for m = 0, 1, ..., 2n − 1 denote the unique cylinder of length n which "contains" the number m expressed as a string of length n in binary. For example, (2)
C0
(2)
= C00 ; C1
(2)
= C01 ; C2
(2)
= C10 ; C3
= C11 .
Notice that Cγ 1 γ 2 ...γ n is same as the interval [γ 1 γ 2 ...γ n 0, γ 1 γ 2 ...γ n 1] := {γ ∈ Ω : γ 1 γ 2 ...γ n 0 ≤ γ ≤ γ 1 γ 2 ...γ n 1}. We need the following observations. By choosing f : Ω → R to be the characteristic function of a cylinder set C ⊂ Ω in Equation 6.18, that is, f (ω) = χC (ω), which we can do because f is continuous, we obtain that for any µ ∈ P(Ω) 1 Z X pi (ω)χC (si (ω))dµ(ω). (U ∗ µ)(C) = i=0 C
We can evaluate this expression in the following four cases. (1) Suppose that C ≤ 01 and C < γ2 = 0γ 1 γ 2 ...: then 1 Z X pi (ω)χC (si (ω))dµ(ω) = p0 µ(S(C)) (U ∗ µ)(C) = i=0 Ω
because p0 (ω)χC (s0 (ω)) = and
½
p0 (ω)χC (0ω) = p0 (ω) = p0 0
when 0ω ∈ C, i.e. ω ∈ S(C); , ω∈ / S(C)
p1 (ω)χC (s1 (ω)) = p1 (ω)χC (1ω) = 0 for all ω ∈ Ω.
(2) Suppose that C ≤ 01 and C > γ2 = 0γ 1 γ 2 ...: then 1 Z X pi (ω)χC (si (ω))dµ(ω) = µ(S(C)) (U ∗ µ)(C) = i=0 Ω
because p0 (ω)χC (s0 (ω)) = and
½
p0 (ω)χC (0ω) = p0 (ω) = 1 when 0ω ∈ C, i.e. ω ∈ S(C); , 0 ω∈ / S(C) p1 (ω)χC (s1 (ω)) = p1 (ω)χC (1ω) = 0.
(3) Suppose that C ≥ 10 and C < γ2 + 12 = 1γ 1 γ 2 ...: then we find 1 Z X ∗ pi (ω)χC (si (ω))dµ(ω) = p1 µ(S(C)) (U µ)(C) = i=0 Ω
because p1 (ω)χC (s1 (ω)) = and
½
p1 (ω)χC (1ω) = p1 (ω) = p1 0
when 1ω ∈ C, i.e. ω ∈ S(C); , ω∈ / S(C)
p0 (ω)χC (s0 (ω)) = p0 (ω)χC (0ω) = 0 for all ω ∈ Ω.
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M ICHAEL BARNSLEY
(4) Suppose that C ≥ 10 and C > γ2 + 12 = 1γ 1 γ 2 ...: then 1 Z X ∗ (U µ)(C) = pi (ω)χC (si (ω))dµ(ω) = 0 i=0 Ω
because p1 (ω)χC (s1 (ω)) = and
½
p1 (ω)χC (1ω) = p1 (ω) = 0 when 1ω ∈ C, i.e. ω ∈ S(C); , 0 ω∈ / S(C)
p0 (ω)χC (s0 (ω)) = p0 (ω)χC (0ω) = 0 for all ω ∈ Ω. Putting the four results together, we obtain: C ≤ 01 and C < γ2 = 0γ 1 γ 2 ... p0 µ(S(C)) µ(S(C)) C ≤ 01 and C > γ2 = 0γ 1 γ 2 ... (6.22) (U ∗ µ)(C) = p1 µ(S(C)) C ≥ 10 and C < γ2 + 12 = 1γ 1 γ 2 ... 0 C ≥ 10 and C > γ2 + 12 = 1γ 1 γ 2 ...
Now let > 0 be given, and let C be any cylinder set that does not contain either of the points γ2 or γ2 + 12 . Then Equation 6.22 implies that for integers n, sufficiently large that γ (mn ) ∈ / C, we have ¯ ∗ ¯ ¯ ¯ ∗ ¯(Uγ µ b)(C) − (Uγ µγ (mn ) )(C)¯ = {either p0 or 1 or p1 or 0} ¯µ b(C) − µγ (mn ) (C)¯ . ¯ ¯ But for sufficiently large n we have ¯µ b(C) − µγ (mn ) (C)¯ < . So for sufficiently large n we have ¯ ∗ ¯ ¯(Uγ µ b)(C) − (Uγ∗ µγ (mn ) )(C)¯ < . It follows that for all sufficiently large n we have ¯ ∗ ¯ ¯(Uγ µ b)(C) − µγ (mn ) (C)¯ < .
It follows that if C is any cylinder set that does not contain either of the points γ2 or γ2 + 12 then (Uγ∗ µ b)(C) = µ b(C). It follows that γ γ 1 (Uγ∗ µ b)(B) = µ b(B) for all B ∈ B(Ω) such that ∈ / B and + ∈ / B. 2 2 2 b({ γ2 + 12 }) = 0. Suppose either µ b({ γ2 }) 6= 0 We now demonstrate that µ b({ γ2 }) = µ or µ b({ γ2 + 12 }) 6= 0. Then we claim that µ b({γ}) 6= 0. For let B = Ω\{ γ2 , γ2 + 12 }. Then ∗ ∗ since (Uγ µ b)(B) = µ b(B) and (Uγ µ b)(Ω) = µ b(Ω) = 1 it follow that (Uγ∗ µ b)({ γ2 , γ2 + 1 b({ γ2 , γ2 + 12 }). But (Uγ∗ µ b)({ γ2 , γ2 + 12 }) = (p0 + p1 )b µ(γ) = µ b({γ}). So it 2 }) = µ follows that µ b({γ}) 6= 0. We now proceed much as in the proof of Theorem 2 to show that µ b({γ}) = 0. Suppose that γ is not eventually periodic. Then we find that µ b({S ◦n γ}) ≥ µ b({γ}) which, since {S ◦n γ}∞ n=1 is an infinite sequence of distinct points, implies µ b(Ω) = ∞ which is wrong. One the other hand suppose that γ is eventually periodic, with eventual period k > 1. Then we find that for some finite positive integer n, µ b({S ◦(n) γ}) < µ b(S ◦(n+k) {γ}) = µ b({S ◦(n) γ}) which is impossible. γ γ 1 ∗ So µ b({ 2 }) = µ b({ 2 + 2 }) = 0. It follows that (Uγ µ b)(B) = µ b(B) for all B ∈ B(Ω), i.e. µ b is indeed an invariant measure as desired. The interesting issue is in relation to uniqueness. First let Pµb (Ω) denote the set of all solutions ρ ∈ P(Ω) of Uγ∗ ρ = ρ such that ρ([0, γ]) = µ b([0, γ]). Then (below) we demonstrate, much as in Theorem 4 that Uγ∗ : Pµb (Ω) → Pµb (Ω) is a contraction
FRACTAL TOPS
31
with respect to dMK . From this it follows that there is at most one solution with prescribed mass on [0, γ] ⊂ Ω. The contraction argument goes as follows. Let µ, υ ∈ P(Ω) be such that µ({ω ∈ Ω : ω < γ}) = υ({ω ∈ Ω : ω < γ}). Then dMK (U ∗ υ, U ∗ µ) Z = max U f (ω)(dν(ω) − dµ(ω)) f ∈Lip1 (Ω)
= =
max
f ∈Lip1 (Ω)
ZΩ Ω
max {
f ∈Lip1 (Ω)
+
Z
f ∈Lip1 (Ω)
+
(p0 f (s0 (ω)) + p1 f (s1 (ω)))(dν(ω) − dµ(ω))
f (s0 (ω))(dν(ω) − dµ(ω))}
max { Z
Z
[0,γ)
[γ,1]
=
(p0 (ω)f (s0 (ω)) + p1 (ω)f (s1 (ω)))(dν(ω) − dµ(ω))
Z
(p0 f (s0 (ω)) + p1 f (s1 (ω)))(dν(ω) − dµ(ω))
[0,γ)
(f (s0 (ω)) + c)(dν(ω) − dµ(ω))}
[γ,1]
where c ∈ R. Now notice that we can choose c so that the function fe : Ω → R defined by ½ p0 f (s0 (ω)) + p1 f (s1 (ω)) ω ∈ [0, γ) e f (ω) = f (s0 (ω)) + c ω ∈ [γ, 1]
is continuous. Then fe ∈ L1 (Ω). It follows that
1 dMK (υ, µ). 2 It follows that there can exist at most one solution with any given mass between zero and one on the interval [0, γ). Now suppose that there are three linearly independent invariant probability measures, solutions to Uγ∗ µ = µ, say µ1 , µ2 , µ3 . Then each of these solutions assigns a different mass to [0, γ] and so we can suppose that dMK (U ∗ υ, U ∗ µ) ≤
µ1 ([0, γ]) < µ2 ([0, γ]) < µ3 ([0, γ]). But then we could construct a linear combination of µ1 and µ3 which has the same mass µ2 on [0, γ], yielding a contradiction. Hence there exists at least one and at most two linearly independent solutions. Hence, via Choquet’s theorem, there are at most two ergodic invariant probability measures. 7. A Random Iteration Algorithm which Stays On Top Here we wiil describe a simple method for random iterating on the top of an IFS.
32
M ICHAEL BARNSLEY
Acknowledgement 1. The author thanks John Hutchinson for many useful discussions and much help with this work. The author thanks Louisa Barnsley for producing the graphics. References [1] M. F. Barnsley, Fractals Everywhere, Academic Press, New York, NY, 1988. [2] M. F. Barnsley and S. Demko, Iterated Function Systems and the Global Construction of Fractals, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 399 (1985), pp. 243-275. [3] M.F.Barnsley and L.F. Barnsley, Fractal Transformations, in "The Colours of Infinity: The Beauty and Power of Fractals", by Ian Stewart et al., Published by Clear Books, London, (2004), pp.66-81. [4] M.F.Barnsley, informal lecture notes entitled Ergodic Theory, Fractal Tops and Colour Stealing (2004). [5] M.F.Barnsley, J.E.Hutchinson, O. Stenflo A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy (2003) to appear in Fractals journal. [6] J. Elton, An Ergodic Theorem for Iterated Maps, Ergodic Theory Dynam. Systems, 7 (1987), pp. 481-488. [7] J. E. Hutchinson, Fractals and Self-Similarity, Indiana. Univ. Math. J., 30 (1981), pp. 713749. [8] N. Lu, Fractal Imaging, Academic Press, (1997). [9] W. Parry, Symbolic Dynamics and Transformations of the Unit Interval, Trans. Amer. Math. Soc., 122 (1966),368-378. [10] A.Rényi, Representations for Real Numbers and Their Ergodic Properties, Acta Math. Acad. Sci. Hung., 8 (1957), pp. 477-493. [11] I. Werner, Ergodic Theorem for Contractive Markov Systems, Nonlinearity 17 (2004) 23032313 Australian National University, Canberra, ACT, Austalia E-mail address :
[email protected]
